Explore forced integrable systems through a 40-minute lecture focusing on the sine-Gordon equation. Delve into the inhomogeneously driven sine-Gordon equation and its applications to tectonic stress transfer and one-dimensional crack dynamics. Examine solutions for kinks and breathers under various boundary conditions, and investigate finite difference methods for heat equations with nonlocal boundary conditions. Learn about the generalized Dirichlet-to-Neumann map for nonlinear evolution PDEs. Gain insights into future research directions and participate in a Q&A session to deepen understanding of these complex mathematical concepts.